A man travels 1 mile due east. Then 5 miles due south, then 2 miles due east and finally 9 miles due north. His displacement is

Options

A. 3 miles
B. 5 miles
C. 4 miles
D. between 5 and 9 miles

Explanation

To find the displacement of the man after he has traveled in various directions, we can break down the problem step by step.

### Step 1: Understand the Movements
The man travels in the following sequence:
1. 1 mile due east
2. 5 miles due south
3. 2 miles due east
4. 9 miles due north

### Step 2: Calculate the Total East-West Movement
- First, he travels 1 mile east.
- Then, he travels another 2 miles east.

Total eastward movement = 1 mile + 2 miles = **3 miles east**.

### Step 3: Calculate the Total North-South Movement
- He travels 5 miles south.
- Then, he travels 9 miles north.

Total northward movement = 9 miles (north) - 5 miles (south) = **4 miles north**.

### Step 4: Determine the Displacement
Now we can visualize the man's final position relative to his starting point. He ends up 3 miles to the east and 4 miles to the north of his starting point.

### Step 5: Use the Pythagorean Theorem
To find the displacement (the straight-line distance from the starting point to the final position), we can use the Pythagorean theorem:

\[ \text{Displacement} = \sqrt{(\text{Eastward Movement})^2 + (\text{Northward Movement})^2} \]

Substituting the values:
\[ \text{Displacement} = \sqrt{(3 \text{ miles})^2 + (4 \text{ miles})^2} \]
\[ = \sqrt{9 + 16} \]
\[ = \sqrt{25} \]
\[ = 5 \text{ miles} \]

### Final Answer
The displacement of the man is **5 miles**.

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